Isometry groups and geodesic foliations of Lorentz manifolds. Part II: Geometry of analytic Lorentz manifolds with large isometry groups

TL;DR

This paper studies Lorentz manifolds with large isometry groups, revealing that non-bi-polarized cases exhibit a geometric rigidity, being locally warped products of constant curvature Lorentz manifolds and Riemannian manifolds.

Contribution

It establishes a geometric rigidity result for non-bi-polarized Lorentz manifolds, showing they are locally warped products, advancing understanding of their structure with large isometry groups.

Findings

Non-bi-polarized Lorentz manifolds are locally warped products.

Rigidity of geometric structure in non-bi-polarized cases.

Extension of previous compactification results to geometric classification.

Abstract

This is Part II of a series on noncompact isometry groups of Lorentz manifolds. We have introduced in Part I, a compactification of these isometry groups, and called ``bi-polarized'' those Lorentz manifolds having a ``trivial '' compactification. Here we show a geometric rigidity of non-bi-polarized Lorentz manifolds; that is, they are (at least locally) warped products of constant curvature Lorentz manifolds by Riemannian manifolds.